"I went into the office of my coaches and said, 'We are manager of the month!' I don't need personal prizes but it's nice now I have another one," Klopp said.
"We are on a good way but there are so many good teams in this league and so many difficult games for us, which we have to be ready for."
Liverpool host sixth-placed Manchester United on Monday and are three points ahead of Jose Mourinho's men in fourth place in the standings.
South Korean Son has stepped up in the absence of injured striker Harry Kane, scoring three league goals in September to help lift his team to second in the table behind Manchester City.
"This is my dream, to get the award. I want to get it a second time, a third time. When I take the award home my family will be crazy like you can't believe. They will be really happy," Son said.
(Reporting by Ian Rodricks in Bengaluru, editing by Ed Osmond)
